Problem 37 - 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. $$\partial M / \partial c$$

Step-by-Step Solution

Answer

The derivative \( \frac{\partial M}{\partial c} \) is \( \frac{B(r - 1)}{(c+r)^2} \) and its sign depends on \( r. \)

Step 1: Identify the Expression for Money Supply

The formula given for the money supply \( M \) in terms of variables \( B, c, \) and \( r \) is \[ M = \frac{c+1}{c+r} B \]. This expression will be used as our starting point for finding the partial derivative with respect to \( c \).

Step 2: Apply the Quotient Rule

The expression for \( M \) is a quotient, so to find \( \frac{\partial M}{\partial c} \), we apply the quotient rule: if \( f(c) = \frac{u(c)}{v(c)} \), then \( f'(c) = \frac{u'(c)v(c) - u(c)v'(c)}{[v(c)]^2} \). In our case, \( u(c) = (c+1)B \) and \( v(c) = c+r \).

Step 3: Differentiate Numerator and Denominator

Find the derivative of the numerator \( u(c) = (c+1)B \) with respect to \( c \), which is \( \frac{d}{dc}[(c+1)B] = B \). Similarly, the derivative of the denominator \( v(c) = c+r \) with respect to \( c \) is \( \frac{d}{dc}[c+r] = 1 \).

Step 4: Use Quotient Rule to Compute Partial Derivative

Substituting the derivatives back into the quotient rule, we have:\[ \frac{\partial M}{\partial c} = \frac{B(c+r) - (c+1)B \cdot 1}{(c+r)^2} = \frac{Bc + Br - Bc - B}{(c+r)^2} \]Simplifying, \[ \frac{\partial M}{\partial c} = \frac{Br - B}{(c+r)^2} = \frac{B(r - 1)}{(c+r)^2} \].

Step 5: Determine the Sign of the Partial Derivative

The sign of \( \frac{\partial M}{\partial c} \) depends on \( r - 1 \). If \( r > 1 \), the derivative is positive, if \( r = 1 \), it is zero, and if \( r < 1 \), it is negative. So, \( \text{sign}( \frac{\partial M}{\partial c} ) \) is determined by \( r \) compared to 1.

Step 6: Explain the Practical Significance

If the fractional reserve ratio \( r \) is greater than 1, it implies that any increase in the cash-to-deposit ratio \( c \) leads to an increase in the money supply \( M \). Conversely, if \( r < 1 \), increasing the cash-to-deposit ratio reduces the money supply. This result emphasizes the role of banking regulations and practices on the availability of money in the economy.

Key Concepts

Money Supply

The term "money supply" refers to the total value of monetary assets available in an economy at a specific time. At its core, money supply includes both cash in circulation and the balances in checking accounts, often referred to as M1 money. There are different measures of money supply, which broaden to include savings accounts and other types of deposits, but M1 is the most liquidity-focused measure.

The money supply equation provided in the problem is \[M = \frac{c+1}{c+r} B\]where:

• \(M\): Money supply
• \(B\): Total cash in the economy
• \(c\): Ratio of cash to checking deposits
• \(r\): Fraction of checking account deposits held as reserves by banks

This formula balances cash and checkable deposits, regulated by how much banks decide to hold back as reserves. In simple terms, changes in either \(c\) or \(r\) impact the broader money supply, effectively altering the amount of money flowing through the economy. Understanding these components is crucial for policymakers aiming to control liquidity and economic stability.

Quotient Rule

The quotient rule is a method for computing the derivative of a quotient of two functions. Specifically, if you have a function \[ f(c) = \frac{u(c)}{v(c)}\]where both \(u(c)\) and \(v(c)\) are differentiable, then the derivative \(f'(c)\) is given by\[f'(c) = \frac{u'(c)v(c) - u(c)v'(c)}{[v(c)]^2}\]In our problem, this rule helps find \[\frac{\partial M}{\partial c}\]by letting \(u(c) = (c+1)B\) and \(v(c) = c+r\).

Following the steps:

• \(u'(c) \) is the derivative of the numerator: since \(u(c) = (c+1)B\), we have \(u'(c) = B\).
• \(v'(c)\) is the derivative of the denominator: \(v(c) = c+r\), thus \(v'(c) = 1\).
• Substitute these into the quotient rule formula to compute the derivative of \(M\) with respect to \(c\).

Applying these steps clearly shows how partial derivatives highlight the influence of one variable on a complex function. This understanding is valuable in economics, where small changes in one factor can affect overall financial dynamics.

Banking Regulations

Banking regulations, particularly regarding reserves, are crucial in determining the economy's money supply. The reserve ratio (\(r\)) defines the proportion of depositors' balances banks must keep on hand. This ratio impacts the amount of money banks can lend out, influencing liquidity and economic activity.

Our partial derivative results show the relationship:

• If \(r > 1\): \(\frac{\partial M}{\partial c} > 0\) any increase in \(c\) raises \(M\).
• If \(r = 1\): \(\frac{\partial M}{\partial c} = 0\), \(M\) remains unchanged.
• If \(r < 1\): \(\frac{\partial M}{\partial c} < 0\), \(M\) decreases as \(c\) increases.

The implications are vast. If regulations increase \(r\) to above 1, banks hold more reserves, thus potentially increasing the economy's money supply with any shift in the cash-to-deposit ratio. Conversely, a lower \(r\) signifies banks can lend more money, which stresses the dependencies between cash preferences and overall liquidity.

These insights guide central banks in setting policies that fine-tune reserve requirements, balancing between too much liquidity, which might fuel inflation, and too little, which can stutter economic growth.