Question

Suppose the FED enlarges the monetary base through open market operations by \(\$ 150\) million: 1) What is the theoretically possible maximum expansion in demand deposits of the total banking system, when the reserve requirement is \(18 \%\) ? 2) What is the expansion in demand deposits and the money supply if we, let \(\Delta \mathrm{H}=\) the amount of the change in the monetary base (reserves), let \(\mathrm{r}=\) the required reserve ratio, and let \(\Delta \mathrm{D}=\) the change in demand deposits taking into consideration all the leakages which the public maintains? The ratio of currency to demand deposits is \(0.30\), the ratio of excess reserves to demand deposits is \(0.125\), the reserve requirement for the time deposits is \(0.04\), and the ratio of savings and time deposits to demand deposits which the public wishes to maintain is \(1.75\).

Short answer

The theoretically possible maximum expansion in demand deposits of the total banking system is \(833.33 \ \text{million}\). The expansion in demand deposits, considering all the leakages maintained by the public, is \(322.5 \ \text{million}\). The expansion in the money supply, considering all the leakages maintained by the public, is also \(322.5 \ \text{million}\).

Step-by-step solution

Part 1: Calculate the Maximum Expansion in Demand Deposits

To calculate the maximum expansion in demand deposits, we use the formula: $$\text{Maximum Expansion in Demand Deposits} = \frac{1}{\text{Reserve Requirement}} \times \text{Change in Monetary Base}$$ Now, we substitute the given values and calculate the maximum expansion in demand deposits.

Plug in the given values

Substitute the values of the reserve requirement, which is \(0.18\) and the change in the monetary base, which is \(150 \ \text{million}\), into the formula. $$\text{Maximum Expansion in Demand Deposits} = \frac{1}{0.18} \times 150 \ \text{million}$$

Calculate the Maximum Expansion in Demand Deposits

Divide \(1\) by \(0.18\) and multiply the result by \(150 \ \text{million}\) to find the maximum expansion in demand deposits. $$\text{Maximum Expansion in Demand Deposits} = 5.56 \times 150 \ \text{million}$$ $$\text{Maximum Expansion in Demand Deposits} = 833.33 \ \text{million}$$ So, the theoretically possible maximum expansion in demand deposits of the total banking system is \(833.33 \ \text{million}\).

Part 2: Calculate the Expansion in Demand Deposits and the Money Supply

We are given the values of: - \(\Delta H = 150 \ \text{million}\) (Change in Monetary Base) - \(r = 0.18\) (Required Reserve Ratio) - \(R_c = 0.30\) (Ratio of Currency to Demand Deposits) - \(R_e = 0.125\) (Ratio of Excess Reserves to Demand Deposits) - \(R_t = 0.04\) (Reserve Requirement for Time Deposits) - \(R_s = 1.75\) (Ratio of Savings and Time Deposits to Demand Deposits) To calculate the expansion in demand deposits and the money supply, we first need to find the money multiplier, using the formula: $$\text{Money Multiplier} = \frac{1 + R_c}{r + R_c + R_e + R_tR_s}$$

Plug in the given values and calculate the Money Multiplier

Substitute the given values into the money multiplier formula: $$\text{Money Multiplier} = \frac{1 + 0.30}{0.18 + 0.30 + 0.125 + 0.04(1.75)}$$ Calculate the money multiplier: $$\text{Money Multiplier} = \frac{1.30}{0.605}$$ $$\text{Money Multiplier} = 2.15$$ Now that we have the money multiplier, we can calculate the change in demand deposits and the money supply using the formula: $$\Delta D = \text{Money Multiplier} \times \Delta H$$

Calculate the Change in Demand Deposits

Substitute the values of the money multiplier and change in the monetary base into the formula: $$\Delta D = 2.15 \times 150 \ \text{million}$$ Calculate the change in demand deposits: $$\Delta D = 322.5 \ \text{million}$$ The expansion in demand deposits, considering all the leakages maintained by the public, is \(322.5 \ \text{million}\).

Calculate the Change in the Money Supply

To get the change in the money supply, we can use the same formula as before: $$\text{Change in Money Supply} = \text{Money Multiplier} \times \Delta H$$ Since we already have these values, we can plug them in: $$\text{Change in Money Supply} = 2.15 \times 150 \ \text{million}$$ Calculate the change in the money supply: $$\text{Change in Money Supply} = 322.5 \ \text{million}$$ The expansion in the money supply, considering all the leakages maintained by the public, is \(322.5 \ \text{million}\).

Key concepts

Money Multiplier

Understanding the money multiplier is key to grasping how banks can lend out more money than they physically have in their vaults. In simplistic terms, it refers to the maximum amount that the money supply can increase based on an initial deposit, after factoring in the reserve requirement.

For example, if a bank receives a deposit of \(100 and the reserve requirement is 10%, the bank can lend out \)90 of that deposit. This $90 can then be deposited in another bank, which can lend out 90% of that amount, and the cycle continues. The process creates money within the economy as the same initial deposit turns into multiple loans and deposits across various banks.

The money multiplier formula, \( M = \frac{1}{reserve \ ratio} \), provides the theoretical maximum for this expansion. However, real-world factors such as cash withdrawals and additional reserve holdings by banks can reduce the actual money multiplier effect.

Demand Deposits

Demand deposits are the balances in bank accounts that depositors can access on demand, meaning without any notice or waiting period. These are typically found in checking accounts, which allow people to use their funds for daily transactions such as paying bills or withdrawing cash.

When a person deposits money into a demand deposit account, the bank can use the majority of these funds to extend loans to other customers. The amount the bank can lend is constrained by the reserve requirement and also determines the potential for demand deposit expansion within the banking system.

Reserve Requirement

The reserve requirement is a regulation set by the central bank that determines the minimum fraction of customer deposits that each bank must keep as reserves, either in their vaults or in an account at the central bank.

The primary purpose of this requirement is to ensure that banks maintain enough liquidity to meet customer withdrawals. It also serves as a tool for monetary policy, influencing the lending capacity of banks. A lower reserve requirement generally means banks can lend more, thereby increasing the money supply. Conversely, a higher reserve requirement restricts lending and thus can tighten the money supply.

Open Market Operations

Open market operations are carried out by central banks to control the money supply and influence interest rates in the economy. This entails the buying and selling of government securities, such as bonds, in the open market.

When a central bank buys securities, they pay out cash, which increases the monetary base—this is generally a reflection of an expansionary monetary policy aimed at reducing interest rates and increasing the money supply. When selling securities, they take in cash, reducing the monetary base and often tightening monetary conditions.

Banking System Liquidity

Liquidity in the banking system refers to the availability of liquid assets—cash and other assets that can be quickly converted to cash—to the banking system as a whole. It reflects the banks’ ability to meet demand for withdrawals and to extend new loans.

Several factors influence banking system liquidity, including central bank policies, the reserve requirement, and the behavior of depositors. Central banks can inject liquidity by decreasing the reserve requirements or through open market operations, buying securities to increase cash in the banking system. High liquidity means more potential for lending and economic growth, while low liquidity can lead to tighter credit conditions and possibly a slowdown in economic activity.