Problem 36 - 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. $$\partial M / \partial r$$

Step-by-Step Solution

Answer

(a) \(-\frac{(c+1)B}{(c+r)^2}\), (b) Negative, (c) Higher reserve ratios decrease money supply.

Step 1: Identify the Function

We are given the formula for the money supply \(M\) as \(M = \frac{c+1}{c+r} B\). To find \(\partial M / \partial r\), we will treat \(c\) and \(B\) as constants and determine how \(M\) changes with respect to \(r\).

Step 2: Apply the Quotient Rule

To find \(\partial M / \partial r\), we use the quotient rule for differentiation, which states \(\frac{d}{dr}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dr} - u \frac{dv}{dr}}{v^2}\). Let \(u = (c+1)B\) and \(v = c+r\). Then, \(\frac{du}{dr} = 0\) and \(\frac{dv}{dr} = 1\).

Step 3: Differentiate using the Quotient Rule

Substitute into the quotient rule: \[ \frac{\partial M}{\partial r} = \frac{(c+r) \cdot 0 - (c+1)B \cdot 1}{(c+r)^2} = \frac{-(c+1)B}{(c+r)^2} \].

Step 4: Determine the Sign of the Derivative

Since \(B\) is positive and both \((c+1)\) and \((c+r)^2\) are positive in a typical economic context, \(\partial M / \partial r\) is negative due to the negative sign in the numerator.

Step 5: Explain the Significance of the Sign

The negative sign of \(\partial M/\partial r\) indicates that as the fraction of checking account deposits that banks hold as cash \(r\) increases, the money supply \(M\) decreases. This suggests that higher reserve ratios reduce the money supply, as less money is circulating within the economy.

Key Concepts

Understanding Money Supply

The money supply in an economy refers to the total amount of cash and checking account balances available at a given time. It plays a critical role in determining economic health and influencing national monetary policies. When considering the money supply, it includes:

• All physical cash in circulation
• Balances held in checking accounts

The relationship between cash (\(B\)), the ratio of cash to checking deposits (\(c\)), and the fraction of checking deposits held by banks as cash reserves (\(r\)) can be encapsulated by the formula for money supply \(M\):\[M = \frac{c+1}{c+r} B\].
This formula indicates how different changes in economic parameters can alter the money supply. For example, increasing \(r\) signifies that banks hold more reserves, which in turn affects \(M\) by reducing the amount of money actively available for transactions in the economy.

The Role of Economics in Money Supply

Economics studies how scarce resources are allocated to meet the needs and desires of people in society. A significant aspect within economics is understanding how money supply influences economic activity, like consumption, investing, and government policy making.

• High money supply: Typically boosts economic activities as more funds are available for spending.
• Low money supply: Might constrain economic activities as resources become tighter.

The indicator provided by \( \partial M / \partial r \), or the partial derivative of the money supply with respect to \( r \), helps economists understand how reserve requirements affect the availability of money. When \( r \) increases, the negative sign of the derivative signals that the overall money supply shrinks. This happens because higher reserves mean less money is available to be lent out or spent, thus impacting economic growth and liquidity.

Utilizing the Quotient Rule in Partial Differentiation

Partial differentiation is a mathematical tool used to analyze how a multivariable function changes when one variable is altered, keeping others constant. In economics, the amount of money circulating, \(M\), is influenced by parameters like \(B\), \(c\), and \(r\), each representing different economic variables.
To differentiate the money supply formula with respect to \(r\), we use the quotient rule. The equation can be expressed as the division of two terms:

• \(u = (c+1)B\)
• \(v = c+r\)

The quotient rule states that for \( \frac{u}{v} \), the derivative is:\[\frac{d}{dr}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dr} - u \cdot \frac{dv}{dr}}{v^2}\]
Applying this to our formula, \( \partial M / \partial r = \frac{-(c+1)B}{(c+r)^2} \).
The role of the quotient rule is crucial in determining how the equilibrium between resources held and available income shifts with changes in economic inputs.