Question

(a) Given the marginal propensity to import \(M^{\prime}(Y)=0.1\) and the information that \(M=20\) when \(Y=0,\) find the import function \(M(Y)\) (b) Civen the marginal propensity to consume \(C^{\prime}(Y)=0.8+0.1 Y^{-1 / 2}\) and the information that \(C=Y\) when \(Y=100\), find the consumption function \(C(Y)\)

Short answer

(a) Import function: \( M(Y) = 0.1Y + 20 \). (b) Consumption function: \( C(Y) = 0.8Y + 0.2 Y^{1/2} + 18 \).

Step-by-step solution

Identify the Integral for Imports

Given that the marginal propensity to import is the derivative of the import function, we have \( M'(Y) = 0.1 \). Therefore, the import function \( M(Y) \) can be found by integrating the derivative: \( M(Y) = \int 0.1 \, dY \).

Integrate to Find the Basic Import Function

Perform the integration: \( M(Y) = 0.1Y + C_1 \), where \( C_1 \) is a constant of integration.

Use Initial Condition to Solve for Constant in Import Function

We know from the problem that \( M = 20 \) when \( Y = 0 \). Substitute these values into the equation: \( 20 = 0.1 \times 0 + C_1 \). Thus, \( C_1 = 20 \). Therefore, the import function is \( M(Y) = 0.1Y + 20 \).

Identify the Integral for Consumption

The marginal propensity to consume is given as \( C'(Y) = 0.8 + 0.1 Y^{-1/2} \). The consumption function \( C(Y) \) can be found by integrating \( C'(Y) \).

Integrate to Find the Basic Consumption Function

Integrate each term in the marginal propensity to consume function: \( C(Y) = \int (0.8 + 0.1 Y^{-1/2}) \, dY = 0.8Y + 0.1 \int Y^{-1/2} \, dY = 0.8Y + 0.2 Y^{1/2} + C_2 \).

Use Initial Condition to Solve for Constant in Consumption Function

We know that \( C(Y) = Y \) when \( Y = 100 \). Therefore, substitute these values into the equation: \( 100 = 0.8 \cdot 100 + 0.2 (100)^{1/2} + C_2 \). Simplifying gives \( 100 = 80 + 2 + C_2 \). Thus, \( C_2 = 18 \). Therefore, the consumption function is \( C(Y) = 0.8Y + 0.2 Y^{1/2} + 18 \).

Key concepts

Marginal Propensity to Import

The concept of "Marginal Propensity to Import" plays a vital role in understanding how a country's import levels change in response to changes in income. Specifically, the marginal propensity to import (MPI) is the fraction of additional income that is spent on importing goods and services from abroad.
In simpler terms, if you earn a bit more money, the MPI tells us how much of that extra cash you will likely spend on imports. For example, in the given exercise, the MPI is 0.1. This means that for every additional dollar earned, 10 cents are spent on imports.
The import function can be derived by integrating the marginal propensity to import. We started with the function's derivative, which was given as 0.1. By integrating, we obtained the import function:

• The basic integral of the derivative gives us: \( M(Y) = 0.1Y + C_1 \).
• Subsequently, applying the initial condition \( M = 20 \) when \( Y = 0 \), enabled us to find that \( C_1 = 20 \).
• Thus, the complete import function becomes \( M(Y) = 0.1Y + 20 \).

By understanding this function, economists can predict import behavior based on income changes.

Consumption Function

The "Consumption Function" reflects the relationship between total consumption and total income in an economy. It helps us understand consumer behavior by showing how consumption changes with income fluctuations.
In the exercise, the marginal propensity to consume is given by a derivative function: \( C^{\prime}(Y) = 0.8 + 0.1 Y^{-1/2} \). This function indicates how consumption changes with each unit change in income.
The consumption function itself can be found by integrating this derivative. Let's break down the steps:

• The first term, \( 0.8 \), integrated, results in \( 0.8Y \).
• The second term, \( 0.1 Y^{-1/2} \), integrated, gives \( 0.2 Y^{1/2} \).
• Adding a constant \( C_2 \), we have the preliminary function: \( C(Y) = 0.8Y + 0.2 Y^{1/2} + C_2 \).

The initial condition given, \( C = Y \) when \( Y = 100 \), helped us solve for \( C_2 \). By substituting and solving, we found \( C_2 = 18 \).
Thus, the final consumption function is \( C(Y) = 0.8Y + 0.2 Y^{1/2} + 18 \). This function is instrumental in understanding how different levels of income can influence consumption.

Integration in Economics

"Integration in Economics" is a mathematical tool used extensively to derive total functions from marginal ones. Within the realm of economics, integration provides crucial insights into aggregate functions like import and consumption.
Integration is the process of calculating the integral of a function, which represents the accumulation of a quantity. When dealing with marginal propensities, economists often integrate to gain the total picture—like total imports or total consumption—which is needed for policy-making and forecasting.
In the given exercise:

• Integration was employed to retrieve the import function from the marginal propensity to import.
• Similarly, it was used for finding the consumption function from the marginal propensity to consume.
• Through the integration of given derivatives, we derived \( M(Y) = 0.1Y + 20 \) and \( C(Y) = 0.8Y + 0.2 Y^{1/2} + 18 \).

Understanding integration helps in visualizing how small, incremental changes—often called marginal changes—aggregate to form significant economic patterns for analysis and decision-making.