Question

Show that an initial deposit of \(\$ 1000\) in a bank can cause an increase in total deposits of the banking system of up to \(\$ 5000 / \mathrm{r}\) dollars, if the required reserve ratio is \(20 \%\).

Short answer

With an initial deposit of \$1000 and a reserve ratio (r) of 20%, the money multiplier (MM) is calculated as \(1/r = 1/0.20 = 5\). By multiplying the initial deposit (D0) by the money multiplier (MM), the total increase in deposits (TD) can be determined: \(TD = D0 \times MM = \$1000 \times 5 = \$5000\). Therefore, the correct statement should be that the total deposits can increase up to \(1000 / \mathrm{r}\) dollars, not \(5000 / \mathrm{r}\) dollars as initially stated.

Step-by-step solution

Understand the Concept of Money Multiplier

A money multiplier is a formula to calculate how an initial deposit can expand the total money supply in the banking system. The formula for money multiplier is \(1 / \mathrm{Reserve Ratio}\). The reserve ratio is the fraction of deposits banks are required to hold as reserves.

Identify given Values

The values given in the task are: Initial deposit (D0): \$1000 Reserve ratio (r): 20% = 0.20

Calculate the Money Multiplier

You can calculate the money multiplier (MM) using the formula: \( \mathrm{MM} = 1 / r \). By plugging in the reserve ratio (r=0.20) into this formula, you get \( \mathrm{MM} = 1 / 0.20 = 5 \).

Calculate the Total Increase in Deposits

Multiply the initial deposit (D0) by the money multiplier (MM) to find out the total increase in the money supply or total deposits (TD): \( \mathrm{TD} = D0 \times MM \). Substituting given values, you get: \( \mathrm{TD} = \$1000 \times 5 = \$5000 \).

Prove the Proposed Statement

The proposed statement is that the total deposits of the banking system can increase up to \(5000 / \mathrm{r}\) dollars. Substituting the value r = 0.20 into that equation gives you \(5000 / 0.20 = \$25000\). This means that for the given value of reserve ratio (r = 20%), the statement is incorrect because the calculated total deposits (\$5000) does not match with the proposed increase (\$25000). It seems that the correct statement should be an increase up to \(1000 / \mathrm{r}\). This would give \(1000 / 0.20 = \$5000\), which match with the calculated total deposit.

Conclusion

The concept of money multiplier is successfully applied in this exercise, however the initial statement that the total deposits of the banking system can increase up to \(5000 / \mathrm{r}\) dollars for initial deposit of \$1000 is not accurate for given reserve ratio of 20%. The correct statement should be that the total deposits can increase up to \(1000 / \mathrm{r}\) dollars.

Key concepts

Reserve Ratio

When we talk about the reserve ratio in the context of banking, we're referring to a regulation that determines the minimum fraction of customer deposits that each bank must keep on hand as reserves, rather than lending out. Think of it as a safety net, ensuring that banks have enough funds available if a large number of customers decide to withdraw their money at the same time.

In our example, the required reserve ratio is 20%, or 0.20 when expressed as a decimal. This means that for every dollar the bank receives in deposits from customers, it needs to keep 20 cents in reserves. The remaining 80 cents can be loaned out to other customers.

This ratio is a crucial piece of the puzzle in understanding how banks can 'create' money through lending and, in this case, why an initial deposit of \(1000 could potentially result in a \)5000 increase in total banking system deposits, based on the money multiplier effect—which we'll explore next.

Banking System Deposits

Now, let's focus on what exactly happens with deposits in the banking system. When a customer makes a deposit, the bank doesn't just put that money in a vault; it puts a part of it to work by lending it to others. This is how banks make most of their money—through the interest on loans.

But here’s the interesting part: when a bank lends money, that money often ends up deposited back into the banking system, creating a cycle of deposits and loans that can greatly magnify the impact of the initial deposit. It sounds like magic, but it's actually the principle of the money multiplier at play. For example, if you follow the steps in the solution, you’ll notice that an initial deposit of \(1000, given a reserve ratio of 20%, will enable the banking system to potentially lend out and thus 'create' a total of \)5000 in bank deposits over the entire process.

Total Money Supply Increase

The total money supply in an economy doesn't only consist of physical money, like cash; it also includes money deposited in banks that can be accessed by checks and digital transactions. This is where the concept of 'money multiplier' really shines—it describes how an initial deposit can ultimately lead to a much larger increase in the total money supply.

As we've calculated, given a 20% reserve ratio, the maximum total increase in deposits, from an initial \(1000, is \)5000. This is an essential concept because it shows the potential for banks to increase the money supply through their lending activities. However, in real-world scenarios, the actual increase might be less due to factors like people holding onto cash or banks choosing to keep reserves above the minimum ratio required. It’s important to remember that while a low reserve ratio can lead to a larger money supply, it can also increase risks for the banking system if too many customers demand their deposits at the same time.