Problem 35 - Applied Calculus

Question

Problems 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

\( \frac{\partial M}{\partial B} = \frac{c+1}{c+r} \) is positive, indicating more cash increases money supply.

Step 1: Identify the function

The money supply is given by the equation \( M = \frac{c+1}{c+r} B \). This function helps us understand how different factors like cash \( B \), cash to deposit ratio \( c \), and the reserve ratio \( r \) impact the total money supply \( M \).

Step 2: Find the partial derivative \( \frac{\partial M}{\partial B} \)

To find the partial derivative of \( M \) with respect to \( B \), we treat \( c \) and \( r \) as constants and differentiate \( M \) with respect to \( B \). This gives:\[ \frac{\partial M}{\partial B} = \frac{c+1}{c+r} \]

Step 3: Determine the sign of \( \frac{\partial M}{\partial B} \)

Since the numerator \( c+1 \) is always greater than 0 and the denominator \( c+r \) is typically also greater than 0 (as both \( c \) and \( r \) are non-negative in practical settings), the fraction \( \frac{c+1}{c+r} \) is positive. This means that \( \frac{\partial M}{\partial B} > 0 \).

Step 4: Explain the significance of the sign

A positive \( \frac{\partial M}{\partial B} \) indicates that as the amount of cash \( B \) increases, the money supply \( M \) also increases, all else being equal. This reflects the intuitive understanding that more cash in the economy contributes to a larger money supply.

Key Concepts

Understanding Money Supply

The term money supply refers to the total amount of money available in an economy at a specific time. It is a crucial measure that economies use to gauge financial health. Money supply encompasses two main components: physical currency like cash, and balances available in checking accounts.
In our discussion, we consider cash represented by \( B \), the cash to checking deposit ratio \( c \), and the reserve ratio \( r \). These factors help us compute the money supply, \( M \), using the formula:

• \( M = \frac{c+1}{c+r} B \)

The primary takeaway from this formula is understanding how changes in cash and these ratios affect the total money supply. This relationship helps policymakers and economists decide whether adjustments in monetary policy might be necessary.
When cash \( B \) increases, the money supply \( M \) generally increases, which can have several effects, including impacts on inflation and interest rates.

The Role of Cash to Checking Deposit Ratio

The cash to checking deposit ratio \( c \) describes the relationship between physical cash and deposits held in checking accounts. This ratio is instrumental in determining the liquidity within an economy. If this ratio is high, it indicates more cash is circulating in hand than is deposited in banks. Conversely, a low ratio suggests higher confidence in depositing money with banks, often leading to more lending and economic activities.
This ratio influences the money supply through the given formula. Since \( c \) appears in both the numerator and denominator of the formula \( \frac{c+1}{c+r} B \), adjustments in this ratio will directly impact the calculation of \( M \). Specifically:

• A higher \( c \) typically results in a lower denominator, increasing the money supply.
• A lower \( c \) boosts the denominator, potentially reducing the money supply.

Understanding the dynamics of this ratio is pivotal for managing liquidity and ensuring the stability of the financial system.

Significant Role of Reserve Ratio

The reserve ratio \( r \) indicates the fraction of deposits that a bank must hold in reserve and not lend out. It acts as a safety measure to ensure banks can meet withdrawal demands. This requirement is set by central banks to control lending and maintain economic stability.
In the context of the money supply formula \( M = \frac{c+1}{c+r} B \), the reserve ratio \( r \) appears in the denominator. This indicates that a higher reserve ratio can decrease the money supply, as it requires banks to hold more funds and limits loan generation. On the other hand, a lower reserve ratio enables more lending, potentially boosting the money supply.
The reserve ratio is a crucial tool in monetary policy. Central banks adjust it to either restrain or encourage economic growth by controlling the amount of money banks can lend. This balance helps in managing inflation and economic activity.