Question

Show that, if the demand for money \(M_{d}\) is a function of the national income \(Y=Y(t)\) and the interest rate \(i=i(t),\) the rate of growth of \(M_{d}\) can be expressed as a weighted sum of \(r_{y}\) and \(r_{i}\) $$r_{M d}=\varepsilon_{M d} y r_{y}+\(\varepsilon_{M_{d}} i f_{i}$$$ where the weights are the elasticities of \)M_{d}\( with respect to \)Y\( and \)i$, respectively.

Short answer

The rate of growth of money demand \( r_{M_d} \) is a weighted sum of \( r_y \) and \( r_i \), with weights corresponding to the elasticities \( \varepsilon_{M_d y} \) and \( \varepsilon_{M_d i} \).

Step-by-step solution

Define the Growth Rates

The rate of growth of any variable is defined as the percentage change in that variable over time. We denote the rate of growth of money demand as \( r_{M_d} \), the rate of growth of national income as \( r_y \), and the rate of growth of the interest rate as \( r_i \).

Express Money Demand Growth in Terms of Elasticities

The elasticities\( \varepsilon_{M_d y} \) and \( \varepsilon_{M_d i} \) represent the responsiveness of money demand \( M_d \) with respect to changes in income \( Y \) and the interest rate \( i \), respectively. These elasticities are mathematically defined as \( \varepsilon_{M_d y} = \frac{\partial M_d / M_d}{\partial Y / Y} \) and \( \varepsilon_{M_d i} = \frac{\partial M_d / M_d}{\partial i / i} \).

Relate Growth Rates Using Elasticities

Substitute the definitions of elasticities into the expression for the growth rate of \( M_d \), yielding the identity:\[ r_{M_d} = \varepsilon_{M_d y} \cdot r_y + \varepsilon_{M_d i} \cdot r_i\]This expression shows how changes in \( Y \) and \( i \) affect the growth rate of \( M_d \) via their respective elasticities.

Key concepts

Money Demand

Money demand refers to the amount of wealth or assets that individuals and businesses choose to hold in the form of money rather than other types of investments or savings. The concept of money demand is critical in monetary economics because it helps us understand how money circulates within an economy. People demand money for several reasons:

• Transaction motive: People need money to conduct day-to-day transactions.
• Precautionary motive: Money is held for unexpected expenses or emergencies.
• Speculative motive: People might hold money to take advantage of future investment opportunities or changes in interest rates.

The demand for money is influenced by factors like income levels and interest rates. Higher national income typically increases money demand as people have more to spend. Conversely, higher interest rates can decrease money demand as people prefer to earn returns through other investments.

Elasticity

Elasticity in economics measures how sensitive a variable is to changes in another variable. When talking about money demand, two primary elasticities are discussed:

• Income elasticity of money demand \(\varepsilon_{M_d y}\): This elasticity measures how responsive the money demand \(M_d\) is to changes in national income \(Y\). If \(\varepsilon_{M_d y}\) is greater than 1, money demand is said to be income elastic, meaning even small changes in income will result in larger changes in money demand.
• Interest rate elasticity of money demand \(\varepsilon_{M_d i}\): This elasticity gauges the responsiveness of money demand to changes in interest rates \(i\). If this elasticity is negative, it implies that higher interest rates tend to reduce money demand as individuals or businesses channel funds into interest-bearing assets.

Understanding elasticity in the context of money demand helps economists and policymakers predict how changes in economic conditions will affect the velocity and volume of money in an economy.

National Income

National income is the total value of goods and services produced by a country over a specific period, usually calculated annually. This measure is crucial because it provides an indication of the economic health of a country.Higher national income typically suggests a higher standard of living, resulting in increased spending and investment. One of the most common measures of national income is the Gross Domestic Product (GDP).National income impacts money demand directly. For instance, as national income \(Y\) increases, individuals and businesses demand more money because:

• Higher income leads to more transactions, increasing the need for money for purchases and investments.
• People may have more flexible spending arrangements, holding more cash for precautionary purposes.

This relationship between national income and money demand is core to understanding how economies adjust to changes in economic activity.

Interest Rate

Interest rates are the price paid for borrowing money or the return earned on savings. They play a critical role in the economy by influencing the money supply and demand balance. Interest rates are determined by several factors, including monetary policy decisions, inflation expectations, and the overall demand for credit. Changes in interest rates can significantly impact money demand:

• When interest rates rise, the opportunity cost of holding money increases, as individuals and businesses could earn more by investing elsewhere. This tends to reduce money demand.
• Conversely, lower interest rates decrease the incentive to save, often increasing money demand as individuals hold more liquid assets.

In macroeconomic models, interest rates are a crucial tool for managing economic growth. Policymakers adjust rates to either cool off an overheating economy or stimulate growth during a downturn. Understanding how interest rates affect money demand is vital for predicting how changes in monetary policy will impact the overall economy.