Problem 34 - Applied Calculus

Question

Are about the money supply, \(M\), which is the total value of all the cash and checking account balances in an economy. It is determined by the value of all the cash, \(B\), the ratio, \(c\), of cash to checking deposits, and the fraction, \(r\), of checking account deposits that banks hold as cash: $$M=\frac{c+1}{c+r} B$$ (a) Find the partial derivative. (b) Give its sign. (c) Explain the significance of the sign in practical terms. $$ \partial M / \partial B $$

Step-by-Step Solution

Answer

The partial derivative \(\frac{\partial M}{\partial B}\) is positive, indicating a direct relationship between cash \(B\) and money supply \(M\).

Step 1: Understanding the Problem

We need to find the partial derivative of the money supply, \(M\), with respect to cash, \(B\), given the formula \(M = \frac{c+1}{c+r} B\). This means we will differentiate \(M\) with respect to \(B\) while treating \(c\) and \(r\) as constants.

Step 2: Differentiate with Respect to B

The expression for \(M\) given is a simple linear function in \(B\). Therefore, take the derivative of \(M\) with respect to \(B\): \(\frac{\partial M}{\partial B} = \frac{c+1}{c+r}\).

Step 3: Determine the Sign of the Partial Derivative

Since \(c\) and \(r\) are both constants and \(c+1\) is always greater than \(c+r\) as long as \(r\) is non-negative, \(\frac{c+1}{c+r}\) will always be positive. Thus, the partial derivative \(\frac{\partial M}{\partial B}\) is positive.

Step 4: Explain the Significance

The positive sign of the partial derivative \(\frac{\partial M}{\partial B}\) signifies that as the total amount of cash, \(B\), increases, the money supply, \(M\), also increases. This indicates a direct relationship between the cash in the economy and the overall money supply.

Key Concepts

money supply

The concept of **money supply** refers to the total amount of money available in an economy at any given time. This includes both the cash held by the public and the funds deposited in checking accounts.
Understanding the money supply is vital, as it relates directly to economic variables such as inflation, interest rates, and economic growth. When talking about money supply, economists often refer to different **measures of money**, such as M1, M2, etc., which capture different components of money.

• **M1**: Includes physical currency and coins, demand deposits, traveler's checks, and other checkable deposits.
• **M2**: Includes all of M1 plus savings deposits, small time deposits, and retail money-market mutual funds.

In our exercise, the formula for money supply, \(M = \frac{c+1}{c+r} B\), highlights that the supply of money is directly proportional to the amount of cash in the economy, \(B\).
The **partial derivative** shows how the money supply changes with a small change in cash, holding other variables constant. A positive derivative here means that increasing cash, \(B\), naturally increases money supply, \(M\). This relationship is simple but crucial for understanding economic policies focused on money management.

cash to checking deposits ratio

The **cash to checking deposits ratio** \(c\) in the formula refers to the proportion of cash that individuals prefer to keep on hand compared to what they deposit into checking accounts.
It's a key parameter as it influences the liquidity preference of consumers. A higher \(c\) suggests people prefer holding cash, potentially limiting the available funds for banking systems to use in loans and other financial operations.

• If \(c\) increases, indicating a preference for more cash, the immediate impact could be reduced funds for banks, affecting interest rates and the bank's ability to lend.
• A lower \(c\), suggests people entrust more money to banks, possibly enhancing the bank's capacity to offer loans and earn interest.

In terms of the money supply formula, the ratio affects how quickly changes in cash flow alter the money supply. It's important to understand how consumer preferences reflected in \(c\) can shape broader economic trends. This ratio can indirectly impact monetary policies driven by central banks seeking to control money supply levels to stabilize an economy.

bank reserves

**Bank reserves** are a crucial part of financial systems, representing the fraction \(r\) of deposits that banks keep as cash on hand rather than lending them out.
These reserves are set based on regulatory requirements to ensure banks can meet withdrawal demands.
The bank reserves are integral to understanding the **money multiplier** effect, where the level of reserves affects how much money banks can create through lending. A higher \(r\) means fewer funds available for banks to loan, potentially slowing economic activity but ensuring more security against bank runs.

• If \(r\) is low, banks can lend more of their deposits, which can spur economic activity through increased investment and consumption.
• Conversely, a high reserve ratio implies safety but limits financial flow in the economy, potentially resulting in slower growth.

In our formula, \(r\) directly influences the scale of money supply through banks' lending activities. As banking regulations and consumer behavior change, understanding \(r\) helps grasp the dynamic between secure banking practices and vibrant economic growth.