Question

In Westlandia, the public holds \(50 \%\) of \(\mathrm{M} 1\) in the form of currency, and the required reserve ratio is \(20 \%\). Estimate how much the money supply will increase in response to a new cash deposit of \(\$ 500\) by completing the accompanying table. (Hint: The first row shows that the bank must hold \(\$ 100\) in minimum reserves \(-20 \%\) of the \(\$ 500\) deposit- against this deposit, leaving \(\$ 400\) in excess reserves that can be loaned out. However, since the public wants to hold \(50 \%\) of the loan in currency, only \(\$ 400 \times 0.5=\$ 200\) of the loan will be deposited in round 2 from the loan granted in round 1.) How does your answer compare to an economy in which the total amount of the loan is deposited in the banking system and the public doesn't hold any of the loan in currency? What does this imply about the relationship between the public's desire for holding currency and the money multiplier?

Short answer

Answer: The money supply increases by $320 in response to a new cash deposit of $500 in Westlandia.

Step-by-step solution

1. Calculate the initial required reserves and excess reserves from the new deposit

First, we need to calculate the amount of required reserves and excess reserves after the cash deposit of \(500. The required reserve ratio is \)20\%\(, so the required reserves will be \)500\times0.20=\$100\(. This leaves \)500-100=\$400$ in excess reserves that can be loaned out.

2. Determine the loan amount that will be deposited in each round

In each round, the bank will lend out the excess reserves and then half of the loan will be deposited back into the banking system because the public wants to hold \(50\%\) of the loan in currency. So we have: Round 1: \(400\times0.5=\$200\) Round 2: \(200\times0.5=\$100\) Round 3: \(100\times0.5=\$50\) and so on...

3. Calculate the increment in the money supply for each round

In each round, the money supply in the banking system increases by the amount of loan that is deposited back in the system: Increment in round 1: \(200\times(1-0.20)=\$160\) Increment in round 2: \(100\times(1-0.20)=\$80\) Increment in round 3: \(50\times(1-0.20)=\$40\) and so on...

4. Calculate the total increase in money supply

Sum the increments for all the rounds to find the total increase in the money supply: Total increase = \(160+80+40+\cdots\) The sum is a geometric series, so we can use the formula for the sum of an infinite geometric series: \(S = \frac{a}{1-r}\), where \(a\) is the first term and \(r\) is the common ratio. Here, \(a=\$160\) and \(r=\frac{1}{2}\), so the total increase in money supply is: \(S = \frac{160}{1-0.5}=\$320\)

5. Comparing with the scenario where the public doesn't hold any of the loan in currency

In this scenario, the entire loan amount would be deposited back into the banking system, and the money multiplier would be higher. If required reserves are still \(20\%\), then the money multiplier would be: \(MM = \frac{1}{0.20}=5\). In this case, the total increase in money supply due to the $500 deposit would be: \(Increase = 500\times (MM-1) = 500\times 4=\$2000\)

6. Implications of public's desire to hold currency on the money multiplier

As the public's desire to hold currency increases, the money multiplier decreases. This means that there is less potential for bank loans to increase the money supply, leading to a slower expansion of the credit in the economy. So, the money supply increases by \(\$320\) in response to a new cash deposit of \(\$500\) in Westlandia. This increase is smaller than the increase in an economy where the public does not hold any of the loan in currency (\(\$2000\)). The public's desire to hold currency has a negative effect on the money multiplier, leading to a slower growth in the money supply and credit expansion in the economy.

Key concepts

Required Reserve Ratio

The required reserve ratio is an essential concept in banking regulations. It determines the percentage of deposits that a bank must keep on hand as reserves without lending them out. This requirement ensures that banks maintain sufficient liquidity to meet customers’ withdrawal demands. In the exercise example, the required reserve ratio is given as 20%. This means that for every deposit, the bank must hold 20% of it as reserves. For instance, with a $500 deposit in Westlandia, the bank is required to keep $100 in reserve. The remaining $400 becomes excess reserves, which the bank can loan out.

This ratio plays a critical role in regulating the money supply within an economy. A lower reserve ratio allows banks to loan out more, increasing the money supply. Conversely, a higher reserve ratio limits how much banks can lend, thus tightening the money supply. Understanding the intricacies of the required reserve ratio helps assess how changes can influence economic growth and stability.

Currency Holding Behavior

Currency holding behavior refers to the tendencies of the public in holding cash versus depositing it in banks. In Westlandia, the public holds 50% of their money as currency. This behavior significantly impacts the money supply and the economy's credit conditions. When people prefer holding onto cash, less money circulates through the banking system, which reduces the amount available for banks to loan.

As demonstrated in the solution, only half of the excess reserves are deposited back into the banking system due to the public holding 50% of loans in currency. Consequently, this behavior diminishes the total potential increase in the money supply. If the public instead deposited all the money back into the banking system, the money supply increase would be substantially more significant. Understanding currency holding behavior offers insights into how public actions can influence the broader economic dynamics.

Geometric Series in Economics

A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero number. In economics, it can represent scenarios where changes follow a patterned progression, like the repetitive lending and re-depositing observed in banking systems.

In the provided exercise, we see the series begin with \(160, following a pattern where each subsequent deposit is halved, showing geometric decay as: \)160, \(80, \)40, and so forth. This scenario forms a geometric series with a first term (\(a\)) of \(160, and a common ratio (\(r\)) of 0.5. The sum of such an infinite geometric series is calculated using the formula: \[ S = \frac{a}{1-r} \]. Here, it results in a total increase of \)320 in the money supply.

Understanding geometric series in this context illustrates how repeated financial activities build upon each other, contributing to the overall economy's growth or contraction. In banking, it explains the cumulative effects of lending and depositing cycles, offering insights into money supply dynamics.